Question

Consider the TTT triple junction illustrated in Figure \(1-38 .\) This triple junction is acceptable because the relative velocity between plates \(\mathrm{C}\) and \(\mathrm{A}, \mathbf{u}_{\mathrm{CA}},\) is parallel to the trench in which plate \(\mathrm{B}\) is being subducted beneath plate \(\mathrm{C}\). The trench between plates \(\mathrm{C}\) and \(\mathrm{B}\) has an azimuth of \(180^{\circ}\) so that \(\mathbf{u}_{\mathrm{CA}}\) has an azimuth of \(0^{\circ}\); assume that \(u_{\mathrm{CA}}=\) \(50 \mathrm{~mm} \mathrm{yr}^{-1}\). Also assume that the azimuth and magnitude of \(\mathbf{u}_{B A}\) are \(315^{\circ}\) and \(60 \mathrm{~mm} \mathrm{yr}^{-1} .\) Determine the azimuth and magnitude of \(\mathbf{u}_{B C}\).

Short answer

Azimuth of \( \mathbf{u}_{\mathrm{BC}} \) is \(80^{\circ}\) and magnitude is approximately \(43 \mathrm{~mm/yr}\).

Step-by-step solution

Define the coordinate system

Set up the coordinate system. Assume \ \( \mathrm{C} \equiv (0,0), \mathrm{A} \equiv (0,50) \), and the velocities are in a Cartesian coordinate system where the positive y-axis is north.

Express velocities in component form

Convert the given velocities into their vector components. For \ \( \mathbf{u}_{\mathrm{CA}} \), the components are \ \((0, 50) \, \mathrm{mm/yr}\). For \ \( \mathbf{u}_{\mathrm{BA}} \) with an azimuth of \ \( 315^{\circ}\), use trigonometry: \ \( u_{x} = 60 \cos(315^{\circ}), \ u_{y} = 60 \sin(315^{\circ})\).

Calculate vector components

For \ \( \mathbf{u}_{\mathrm{BA}} \), calculate \ \( x = 60 \times \cos(315^{\circ}) = 60 \times \frac{1}{\sqrt{2}} = 42.43 \, \mathrm{mm/yr}, \, y = 60 \times \sin(315^{\circ}) = 60 \times \frac{-1}{\sqrt{2}} = -42.43 \, \mathrm{mm/yr} \).

Apply the vector addition equation

Use the vector addition principle \ \( \mathbf{u}_{\mathrm{BC}} = \mathbf{u}_{\mathrm{BA}} + \mathbf{u}_{\mathrm{CA}} \). Substituting the vector components: \ \( (x_{BC}, y_{BC}) = (42.43, -42.43) + (0, 50) \).

Solve for \(\mathbf{u}_{\mathrm{BC}} \) components

Combine the vector components: \ \( x_{BC} = 42.43 \mathrm{~mm/yr}, \, y_{BC} = -42.43 + 50 = 7.57\mathrm{~mm/yr} \). So \ \( \mathbf{u}_{\mathrm{BC}} = (42.43, 7.57) \).

Calculate magnitude of \( \mathbf{u}_{\mathrm{BC}} \)

Compute the magnitude using the formula: \ \( \|\mathbf{u}_{BC}\| = \sqrt{42.43^2 + 7.57^2} \). Evaluating gives approximately 43 \, \mathrm{mm/yr}.

Calculate azimuth for \( \mathbf{u}_{\mathrm{BC}} \)

Use the tangent function to find the azimuth: \ \( \theta = \tan^{-1}\left(\frac{7.57}{42.43}\right) \). Solving gives approximately \(10^{\circ}\), which represents an azimuth of \ \( \tan^{-1}(\frac{y_{BC}}{x_{BC}}) \rightarrow 10^{\circ} \text{ from east} \). Adjust to standard azimuth: \ 80^{\circ}.

Key concepts

Plate Tectonics

Plate tectonics is a scientific theory explaining the movement of the Earth's lithosphere, which is divided into large plates. These plates float on the semi-fluid asthenosphere beneath, and their interactions cause phenomena like earthquakes, volcanic activity, and the formation of mountain ranges.
A triple junction is a point where three tectonic plates meet, and it is a significant feature in studying plate dynamics. These junctions can be classified into different types based on plate boundaries: Transform fault (T), Rift (R), or Subduction zone (S) leading to combinations like TTT (three transform faults) or TTR (two transform faults and one rift).
In the given exercise, we look at a TTT triple junction which involves three plates meeting where the relative velocity between some of them needs to be understood to determine the type of boundary connections they emulate.

Vector Addition

Vector addition is a fundamental concept in physics and mathematics where two or more vectors are combined to determine a resultant vector.
Vectors have both direction and magnitude, which can be added using a graphical or an algebraic method. In Cartesian coordinates, this involves summing the respective components (i.e., x and y coordinates) of the vectors.
In the exercise, the relative plate velocity vectors, such as \( \mathbf{u}_{\mathrm{CA}} \) and \( \mathbf{u}_{\mathrm{BA}} \), are added to find \( \mathbf{u}_{\mathrm{BC}} \).

• To add vectors algebraically, each vector is split into its component form based on the defined coordinate system measures.
• Then, the components of each vector in the same direction are added.
• This results in a vector that gives the combined effect of the original vectors.

Understanding how to correctly add vectors is crucial when evaluating dynamic systems like tectonic plates because it allows for accurate predictions of movement.

Coordinate Systems

A coordinate system is a framework used to identify the position of points in space, which can be two-dimensional or three-dimensional. The Cartesian coordinate system is commonly used due to its simplicity and ease of use.
In a Cartesian system:

• The x-axis runs horizontally and the y-axis runs vertically in a two-dimensional setup. North is usually assigned to the positive y-axis.
• Each point is described by an ordered pair (x, y).
• Applications of such systems help in determining positions and velocities in real-world scenarios like plate tectonics.

For the exercise, the coordinate system is set up with specific reference points to describe the movement of tectonic plates accurately. Each point and subsequent vector can be used to calculate velocities between various plates.

Azimuth Calculation

Azimuth is an angular measurement in a spherical coordinate system, such as the one used in compass navigation. Typically, the azimuth is measured clockwise from a reference direction, usually north, which is set at 0 degrees.
Calculating the azimuth involves using trigonometry, often the inverse tangent function, formulated as \[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \] where \( y \) and \( x \) are the coordinates of the point of interest.
In the given example, the azimuth for \( \mathbf{u}_{\mathrm{BC}} \) is calculated through its components, resulting in an angle that represents its orientation from the north. The calculation adjusts to azimuth angles by mapping typical mathematical angles into navigational counterparts.

• This is crucial in understanding directions on a spherical body like Earth.

Such knowledge helps interpret how and where tectonic plates move over the globe, affecting global geological phenomena.